Multi-criteria decision making approach based on PROMETHEE with probabilistic simplified neutrosophic sets

Abstract

Probabilistic simplified neutrosophic set $\left( {PSNS} \right)$ is an important tool to describe the vagueness existing in the real life. In this study, we define a PSNS and discuss some of theoretical set operations of $PSNSs$. Also we propose the concepts of module on $PSNSs$, as well as inner product and projection operator between two $PSNSs$. In relation to this new set, we introduce a probabilistic simplified neutrosophic number $\left( {PSNN} \right)$. A $PSNN$ has three components which is called probabilistic-valued truth membership degree, probabilistic-valued indeterminacy membership degree and probabilistic-valued falsity membership degree, respectively. We give some of algebraic operational rules, a score function and an accuracy function on $PSNNs$. Furthermore, we introduce two aggregation operators called the probabilistic simplified neutrosophic weighted arithmetic average operator and the probabilistic simplified weighted geometric average operator on $PSNNs$. Furthermore, we determine weights of criteria with a method that is based on fuzzy measure and develop a method based on preference function to determine weight of each decision maker. We present an extended PROMETHEE method based on $PSNSs$ for group decision problems. Finally, as an application of this theory, we give a practice on a multi-criteria group decision making problem based on $PSNNs$ by using extended PROMETHEE method to ensure stability of the proposed method.